Perfect digit-to-digit invariant

A perfect digit-to-digit invariant (PDDI) (also known as a Munchausen number) is a natural number that is equal to the sum of its digits each raised to a power equal to the digit.

$n=d_{k}^{d_{k}}+d_{k-1}^{d_{k-1}}+\dots +d_{2}^{d_{2}}+d_{1}^{d_{1}}\,.$

0 and 1 are PDDIs in any base (using the convention that 0⁰ = 0). Apart from 0 and 1 there are only two other PDDIs in the decimal system, 3435 and 438579088 (sequence A046253 in the OEIS). Note that the second of these is only a PDDI under the convention that 0⁰ = 0, but this is standard usage in this area.

$3^{3}+4^{4}+3^{3}+5^{5}=27+256+27+3125=3435$

$4^{4}+3^{3}+8^{8}+5^{5}+7^{7}+9^{9}+0^{0}+8^{8}+8^{8}$

$=256+27+16777216+3125+823543+387420489+0+16777216+16777216=438579088$

More generally, there are finitely many PDDIs in any base. This can be proved as follows:

Let $b$ be a base. Every PDDI $n$ in base $b$ is equal to the sum of its digits each raised to a power equal to the digit. This sum is less than or equal to $a(b-1)^{b-1}$, where $a$ is the number of digits in $n$, because $b-1$ is the largest possible digit in base $b$. Thus,

$a(b-1)^{b-1}\geq n\geq b^{a-1}.$

The expression $a(b-1)^{b-1}$ increases linearly with respect to $a$, whereas the expression $b^{a-1}$ increases exponentially with respect to $a$. So there is some $k>0$ such that

$\forall a\geq k,\,\,a(b-1)^{b-1}<b^{a-1}.$

There are finitely many natural numbers $n$ with fewer than k digits, so there are finitely many natural numbers $n$ satisfying the first inequality. Thus, there are only finitely many PDDIs in base $b$.

In all bases 1 is a PDDI.
In base 3 there are 2 PDDI's, namely 12 and 22. (5 and 8 in decimals)
In base 4 there are 2 PDDI's, namely 131 and 313. (29 and 55 in decimals)
In base 6 there are 2 PDDI's, namely 22352 and 23452. (3164 and 3416 in decimals)
In base 7 there is 1 PDDI's, namely 13454. (3665 in decimals)
In base 9 there are 3 PDDI's, namely 31, 156262 and 1656547. (28, 96446 and 923362 in decimals)

When the convention $0^{0}=0$ is used the following numbers are also PDDI's

In all bases 0 is a PDDI.
In base 4 there is one additional PDDI, namely 130. (28 in decimal)
In base 5 there are 2 PDDI's, namely 103 and 2024. (28 and 264 in decimal)
In base 8 there are 2 PDDI's, namely 400 and 401. (256 and 257 in decimal)
In base 9 there are 3 additional PDDI's, namely 30, 1647063 and 34664084. (27, 917139 and 16871323 in decimal)

External links

• Parker, Matt. "3435 – Matt's Ex". Numberphile. Brady Haran.